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Special Products · Grade 9 · Worksheet 3

1. An engineer is designing a solar panel array that will be installed on a rectangular section of a rooftop. The array's dimensions are represented by (5x + 8) meters in length and (5x - 8) meters in width. To calculate the total power output, she needs to determine the area of the array. What is the simplified expression for the area of the solar panel array in square meters? Answer: 25x² - 64 Solution: The difference of squares is a special algebraic pattern where multiplying two binomials with the same terms but opposite signs results in the square of the first term minus the square of the second term.
   Full step-by-step solution

   The difference of squares is a special algebraic pattern where multiplying two binomials with the same terms but opposite signs results in the square of the first term minus the square of the second term. For example, (3y + 4)(3y - 4) would simplify to 9y² - 16. This pattern appears frequently in engineering, physics, and architecture when dealing with rectangular areas where dimensions differ by a constant amount.

2. (9x² - 16y⁴) = ? Answer: (3x - 4y²)(3x + 4y²) Solution: Recognize the pattern a² - b² = (a - b)(a + b) Identify a² = 9x², so a = 3x Identify b² = 16y⁴, so b = 4y² Apply the difference of squares formula: (3x - 4y²)(3x + 4y²) The answer is (3x - 4y²)(3x + 4y²).
   Full step-by-step solution

   Step 1: Recognize the pattern a² - b² = (a - b)(a + b) Step 2: Identify a² = 9x², so a = 3x Step 3: Identify b² = 16y⁴, so b = 4y² Step 4: Apply the difference of squares formula: (3x - 4y²)(3x + 4y²) The answer is (3x - 4y²)(3x + 4y²).

3. A rectangular prism has dimensions (2x + 3) cm by (2x - 3) cm by (x + 4) cm. Using the difference of squares pattern, find the volume of the prism as a simplified polynomial expression in terms of x. Answer: 4x^3 + 7x^2 - 36x - 63 Solution: The volume of a rectangular prism is length × width × height. First, multiply the length and width: (2x + 3)(2x - 3) This is a difference of squares: (a + b)(a - b) = a² - b² So (2x + 3)(2x - 3) = (2x)² - (3)² = 4x² - 9 Now multiply this result by the height: (4x² - 9)(x + 4) Use the…
   Full step-by-step solution

   Step 1: The volume of a rectangular prism is length × width × height. Step 2: First, multiply the length and width: (2x + 3)(2x - 3) Step 3: This is a difference of squares: (a + b)(a - b) = a² - b² Step 4: So (2x + 3)(2x - 3) = (2x)² - (3)² = 4x² - 9 Step 5: Now multiply this result by the height: (4x² - 9)(x + 4) Step 6: Use the distributive property: 4x²(x + 4) - 9(x + 4) Step 7: = 4x³ + 16x² - 9x - 36 Step 8: The volume is 4x³ + 16x² - 9x - 36 cubic centimeters.

4. (14x³ + 11y²)(14x³ - 11y²) = ? Answer: 196x⁶ - 121y⁴ Solution: Recognize this is a difference of squares: (a + b)(a - b) = a² - b² Identify a = 14x³ and b = 11y² Square the first term: (14x³)² = 14² × (x³)² = 196 × x⁶ = 196x⁶ Square the second term: (11y²)² = 11² × (y²)² = 121 × y⁴ = 121y⁴ Apply the formula: a² - b² = 196x⁶ - 121y⁴ The answer is 196x⁶ - 121y⁴.
   Full step-by-step solution

   Step 1: Recognize this is a difference of squares: (a + b)(a - b) = a² - b² Step 2: Identify a = 14x³ and b = 11y² Step 3: Square the first term: (14x³)² = 14² × (x³)² = 196 × x⁶ = 196x⁶ Step 4: Square the second term: (11y²)² = 11² × (y²)² = 121 × y⁴ = 121y⁴ Step 5: Apply the formula: a² - b² = 196x⁶ - 121y⁴ The answer is 196x⁶ - 121y⁴.

5. An architect is designing a rectangular conference room with dimensions (5x + 8) feet by (5x - 8) feet. The client wants to install custom carpeting and needs to know the exact area of the room. What is the simplified expression for the area of the conference room in square feet? Answer: 25x^2 - 64 Solution: Area = (5x + 8)(5x - 8) Apply the difference of squares formula: (a + b)(a - b) = a² - b² Here, a = 5x and b = 8 Area = (5x)² - (8)² Area = 25x² - 64 The simplified expression for the area is 25x² - 64 square feet.
   Full step-by-step solution

   Step 1: The area of a rectangle is length × width Step 2: Area = (5x + 8)(5x - 8) Step 3: Apply the difference of squares formula: (a + b)(a - b) = a² - b² Step 4: Here, a = 5x and b = 8 Step 5: Area = (5x)² - (8)² Step 6: Area = 25x² - 64 The simplified expression for the area is 25x² - 64 square feet.

6. A rectangular garden is divided into two sections by a diagonal path. The garden has a length of (6x + 1) meters and a width of (6x - 1) meters. A square flower bed with side length (x + 4) meters is placed in one corner of the garden. Noah needs to find the area of the remaining garden space. Express your answer as a simplified polynomial in terms of x. Answer: 35x^2 - 8x - 17 Solution: Find the total area of the rectangular garden using the difference of squares pattern. The length is (6x + 1) and the width is (6x - 1). Total area = (6x)^2 - (1)^2 = 36x^2 - 1 square meters.
   Full step-by-step solution

   Step 1: Find the total area of the rectangular garden using the difference of squares pattern. The length is (6x + 1) and the width is (6x - 1). Step 2: Apply the formula (a+b)(a-b) = a^2 - b^2, where a = 6x and b = 1. Step 3: Total area = (6x)^2 - (1)^2 = 36x^2 - 1 square meters. Step 4: Find the area of the square flower bed. Side length = (x + 4), so area = (x + 4)^2. Step 5: Expand (x + 4)^2 using the perfect square formula: (x + 4)^2 = x^2 + 8x + 16. Step 6: Subtract the flower bed area from the total garden area: (36x^2 - 1) - (x^2 + 8x + 16). Step 7: Distribute the negative sign: 36x^2 - 1 - x^2 - 8x - 16. Step 8: Combine like terms: (36x^2 - x^2) = 35x^2, (-8x) stays, (-1 - 16) = -17. Step 9: The remaining area is 35x^2 - 8x - 17 square meters. The answer is 35x^2 - 8x - 17.

7. (12x² + 7y)(12x² - 7y) = ? Answer: 144x⁴ - 49y² Solution: Recognize this is a difference of squares: (a + b)(a - b) = a² - b² Identify a = 12x² and b = 7y Square the first term: (12x²)² = 144x⁴ Square the second term: (7y)² = 49y² Apply the formula: a² - b² = 144x⁴ - 49y² The answer is 144x⁴ - 49y².
   Full step-by-step solution

   Step 1: Recognize this is a difference of squares: (a + b)(a - b) = a² - b² Step 2: Identify a = 12x² and b = 7y Step 3: Square the first term: (12x²)² = 144x⁴ Step 4: Square the second term: (7y)² = 49y² Step 5: Apply the formula: a² - b² = 144x⁴ - 49y² The answer is 144x⁴ - 49y².

8. Mason is designing a rectangular metal plate for a structural support beam. The length of the plate is (7x + 9) centimeters and the width is (7x - 9) centimeters. To calculate the amount of material needed, he must find the area of the plate. What is the simplified expression for the area of the metal plate in square centimeters? Answer: 49x² - 81 Solution: Area of a rectangle = length × width, so Area = (7x + 9)(7x - 9). Recognize this matches the difference of squares pattern: (a + b)(a - b) = a² - b², where a = 7x and b = 9.
   Full step-by-step solution

   Step 1: Area of a rectangle = length × width, so Area = (7x + 9)(7x - 9). Step 2: Recognize this matches the difference of squares pattern: (a + b)(a - b) = a² - b², where a = 7x and b = 9. Step 3: Apply the formula: (7x)² - (9)². Step 4: Calculate each square: (7x)² = 49x² and (9)² = 81. Step 5: Subtract: 49x² - 81. The simplified expression for the area is 49x² - 81 square centimeters.